OPTIMAL DESIGN OF PHOTONIC CRYSTALS MEN HAN (B.Eng., NUS, S.M., MIT, S.M., NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN COMPUTATIONAL ENGINEERING SINGAPORE-MIT ALLIANCE NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgments Throughout the course of this doctoral research, many great minds and kind souls have been part of the influencing and shaping force. This has been a most memorable journey filled with their wisdom, advice and friendship. My deepest gratitude goes to Professor Jaime Peraire, a truly illuminating advisor and a tremendous source of inspiration, motivation and guidance. Ever since our first encounter at a faculty-student meeting in Singapore back in my college senior year in 2005, I have always been fascinated by Jaime’s brilliant scientific ideas, his unstoppable energy and captivating humor. I am very grateful for his supportive and inspirational mentoring style, which has fostered me to become more independent and self-initiative. My equally sincere gratitude goes to Professor Lim Kian Men. He has been the most understanding and accommodating advisor, under whom I was granted tremendous trust to explore and experiment freely. At the same time, he is always available to provide any help and advice when needed which I deeply respect and appreciate. I would like to thank my thesis committee members. Professor Toh Kim Chuan has been one of my deeply revered professors, not only because of his kind and helpful advice, but also for his efficient and remarkably (almost) bugfree “SDPT3” solver that has made my life so much easier. Professor Karen Willcox has never been shy to share with me her personal experiences and timely give me the much-needed encouragement. She has always inspired me with her perseverant willpower and ingenuous personality. For that, I consider Karen my role model. I am also very grateful to all my thesis committee members as well as advisors for being accommodating to overcome the time difference, which has allowed me to receive constant advice via video conferences until finally defend this work. I would also like to thank my collaborators with whom two journal papers have been published on this project. Doctor Ngoc-Cuong Nguyen is in fact more than a collaborator, he has been my closest and most respectable mentor for the past five years. I not think I could have learned and grown so much as i a researcher if it had been for his constant encouragement and endless advice. His own dedication to research and the rewarding accomplishments have greatly inspired me to follow in his footsteps closely. Professor Robert Freund is anther one of the professors that I constantly look up and revere. I commend him for his respectable work ethics and thank him for his unwearied guidance. Great appreciation goes to Professor Pablo Parillo for his flashes of genius that have enabled us to overcome many obstacles in this research. Special thanks go to Professor Steven Johnson for his most generous advice and helpful suggestions that have enlightened me. Numerous staff in SMA offices from both Singapore and MIT have lent a helpful hand. I would like to thank, Jason Chong for working on a tight schedule to make the oral defense happen on time, as well as Juliana and Neo for their timely assistance. Thanks to Jocelyn Sales and Doctor John Desforge for the tender loving care they promised even before the start of my doctoral study. They have made my journey with SMA so warm, fun and memorable. Jean from ACDL and Laura from CDO have been nothing short of kindness, they have painted beautiful colors to my cold Boston days. I would like to extend a warm thank you to all my friends who have been helpful and supportive throughout the journey of my doctoral study. The precious moments we shared across the kitchen counters, over the dinner tables, at celebratory parties and gatherings, have not only helped me keep my sanity after the long days of work, but have also been an immensely enjoyable part of my life. In no particular order, many thanks go to, Chewhooi, Haiying and WW, Christina, Sunwei, Julei, Josephine, Shann, Zhoupeng, Ruxandra, Smaranda, Alex, Thanh, and Vanbo. Finally, none of this could have been possible without the unwavering love and care of my family. My parents have been the best parents a child can ever have. They have always supported every dream I wish to pursue and created a wonderful and comforting life for me to completely immerse myself in. Their unconditional love and constant faith have never been absent despite the distance. To them, I give my heart-felt gratitude. The most special thank-yous go to my fiance, Bogdan Fedeles, who has been ii a constant source of love, inspiration and strength all these years. I am deeply grateful for his passion and appreciation, for his patience and tolerance, for his encouragement and support. Bogdan is also my best friend in life, with whom I can have engaging conversations on myriad topics, with whom I can play a competitive board game or tennis match, with whom I can enjoy a home-cooked dinner or a relaxing movie, with whom I can have a whole lifetime of happiness. Thank you, Bogdan, for being my life and for loving me. To my family, I dedicate this thesis. iii Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables ix List of Figures xiii Introduction 1.1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Photonic crystals . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Optimal design . . . . . . . . . . . . . . . . . . . . . . . . Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Thesis contributions . . . . . . . . . . . . . . . . . . . . . 1.2.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . 10 Building Blocks 2.1 2.2 2.3 2.4 12 Review of Electromagnetism in Dielectric Media . . . . . . . . . 12 2.1.1 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Symmetries and Bloch-Floquet theorem . . . . . . . . . . 15 Review of Functional Analysis . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Linear and bilinear functionals . . . . . . . . . . . . . . . 20 Review of Finite Element Method . . . . . . . . . . . . . . . . . 21 2.3.1 Variational or weak formulation . . . . . . . . . . . . . . . 21 2.3.2 Spaces and basis . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Discrete equations . . . . . . . . . . . . . . . . . . . . . . 23 Review of Convex Optimization . . . . . . . . . . . . . . . . . . . 24 iv 2.4.1 Convex sets and cones . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Semidefinite program . . . . . . . . . . . . . . . . . . . . 27 2.4.3 Second-order cone program . . . . . . . . . . . . . . . . . 27 2.4.4 Linear fractional program . . . . . . . . . . . . . . . . . . 27 Band Structure Calculation 3.1 3.2 3.3 29 Band Structure of Two-dimensional Photonic Crystal Structure . 31 3.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . 31 3.1.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.3 Mesh refinement . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.4 Results and discussion . . . . . . . . . . . . . . . . . . . . 39 Band Structure of Three-dimensional Photonic Crystal Fiber . . 44 3.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . 56 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Bandgap Optimization of Photonic Crystal Structures 65 4.1 The Band Gap Optimization Problem . . . . . . . . . . . . . . . 65 4.2 Band Structure Optimization . . . . . . . . . . . . . . . . . . . . 66 4.2.1 4.3 Reformulation of the band gap optimization problem using subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2.2 Subspace approximation and reduction . . . . . . . . . . . 68 4.2.3 Fractional SDP formulations for TE and TM polarizations 71 4.2.4 Multiple band gaps optimization formulation . . . . . . . 74 4.2.5 Computational procedure with mesh adaptivity . . . . . . 76 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.2 Choices of parameters . . . . . . . . . . . . . . . . . . . . 79 4.3.3 Computational cost . . . . . . . . . . . . . . . . . . . . . 81 4.3.4 Mesh adaptivity . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.5 Optimal structures with single band gap . . . . . . . . . . 90 4.3.6 Optimal structures with multiple band gaps . . . . . . . . 95 v 4.3.7 4.4 Optimal structures with complete band gaps . . . . . . . 101 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Single-Polarization Single-Mode Photonic Crystal Fiber 5.1 5.2 5.3 105 The Optimal Design Problem . . . . . . . . . . . . . . . . . . . . 106 5.1.1 Formulation I . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1.2 Formulation II . . . . . . . . . . . . . . . . . . . . . . . . 114 5.1.3 Trust region . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.2 Optimal structures . . . . . . . . . . . . . . . . . . . . . . 124 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Conclusions 134 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Bibliography 140 vi Summary The present work considers the optimal design of photonic crystals. Convex optimization will be formally used for the purpose of designing photonic crystal devices with desired eigenband structures. In particular, two types of devices will be studied. The first type is a “two-dimensional” photonic crystal with discrete translational symmetry in the transverse plane, and is invariant along the longitudinal direction. The desired band structure of this device is one with optimal band gap between two consecutive eigenmodes. The second type is a three-dimensional photonic crystal fiber, which can be constructed schematically from the first type of device by introducing a defect in the transverse plane and breaking the translational symmetry. The desired feature of this device is to possess a band structure with an optimal band width at a certain propagation constant. The two design problems are analogous in that the difference between two consecutive eigenmodes is the objective function, and the disparity lies in the evaluation of eigenvalues with respect to different sets of wave vectors. The mathematical formulations of both optimization problems lead to an infinite-dimensional Hermitian eigenvalue optimization problem parameterized by the dielectric function. To make the problem tractable, the original eigenvalue problem is discretized using the finite element method into a series of finite-dimensional eigenvalue problems for appropriate values of the wave vector parameter. The resulting optimization problem is large-scale and non-convex, with low regularity and a non-differentiable objective. By restricting to appropriate sub-eigenspaces, and employing mesh adaptivity, we reduce the large-scale non-convex optimization problem via reparametrization to a sequence of smallscale convex semidefinite programs (SDPs) for which modern SDP solvers can be efficiently applied. We present comprehensive optimal structures of photonic crystals of different lattice types with numerous single and multiple, absolute and complete optimal band gaps, as well as single-mode single-polarization photonic crystal fiber structures of different lattice types with optimal band width for which only single guided mode can propagate. vii The optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design. viii List of Tables 4.1 Computational cost for single band gap optimization in square lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Average computational cost (and the breakdown) of 10 runs as the mesh is refined uniformly. . . . . . . . . . . . . . . . . . . . . 4.3 82 84 Comparison of computational cost between uniform and adaptive meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 ix For notational convenience, we rewrite the system of equations in the following operator form: Au = λMu, in Ωs , (3.2.5) where λ ≡ (ω/c)2 , and   ∇⊥ × A(ε, β) ≡  ε−1 ∇ ⊥× + iβ∇⊥ · ε−1 β ε−1 iβε−1 ∇  ⊥ −∇⊥ · ε−1 ∇⊥  ,  M ≡ I,   H⊥  u≡ . Hz (3.2.6) We denote by (um , λm ) the m-th pair of eigenfunction and eigenvalue of (3.1.3) and assume that these eigen pairs are numbered in ascending order: < λ1 ≤ λ2 ≤ · · · ≤ λ∞ . The formulation shown in (3.2.6) is also known as the resonance problem. When the third dimension is homogenous as the photonic crystal fiber, the original three-dimensional vectorial problem (3.1) is reduced to a two-dimensional vectorial eigenproblem; comparing with the eigenproblem in section 3.1, the original two-dimensional vectorial problem was reduced to two two-dimensional scalar problems, when the propagation wave vector k(x, y) is limited to the cross section of the two-dimensional photonic crystal structures. We need to keep our numerical scheme simple enough for the optimization formulation (to be discussed in Chapter 5) yet not oversimplified to result in the loss of accuracy. Essentially, two types of eigenmodes will be sought. The first type relates to the modes in a photonic crystal cladding with perfect translational symmetry, from which the fundamental space filling mode will be calculated. A periodic boundary condition on the primitive unit cell Ω suffices in this case. The second type involves the modes in a disrupted cladding, also known as the photonic crystal fiber, or in the most general case, the waveguide. In this case, we are seeking only the guided modes, whose corresponding magnetic field decays exponentially away from the core to zero inside the cladding region within a finite domain, i.e., the super cell Ωs if it is set large enough. While most boundary conditions would be satisfied in this case, the periodic boundary condition is again chosen for consistency and simplicity. 49 3.2.2 Discretization Mixed continuous Galerkin The most well-known issue with the finite element method for solving threedimensional electromagnetic eigenproblems is the introduction of spurious modes [54], i.e., nonphysical eigenvalues which appear in the computed spectrum polluting the correct, physical eigenvalues. This is believed to attribute to the improper modeling of the discontinuities of the field variables across dielectric boundaries by node elements. Recall that node elements enforce the continuity of the field variables over the entire computation domain. However, Maxwell equations require the continuity of the tangential component of the field variables across dielectric boundaries. Tangential vector finite elements, also known as edge elements, need to be introduced to enforce the tangential continuity of the elements. In our optical waveguide problem where the dimensional- ity is reduced to two due to longitudinal symmetry, the longitudinal component of the field variable is continuous across material boundaries, while the transverse components are discontinuous and the continuity of their tangential components need to be enforced. As a result, the longitudinal field variable uz (∈ H (Ω)) is represented by node elements, and the transverse field variable u⊥ (∈ H(curl, Ω) ≡ v ∈ [L2 (Ω)]2 | ∇⊥ × v ∈ [L2 (Ω)]2 ) is represented by edge elements to ensure that all the non-zero eigensolutions correspond to valid waveguide modes [37, 22]. The strategy described above is called the mixed formulation, and was first proposed by Kikuchi in [35]. In the same paper, he also pointed out the presence of an infinitely degenerate eigenvalue “0” whose eigenspace lies in the functional space of the field variables. To understand this issue, we can apply the Helmholtz decomposition to split the field variable into curl-free and divergence-free parts:        H⊥   ∇⊥ ϕ   u⊥  .  = + uz Hz iβϕ curl−f ree (3.2.7) div−f ree The curl-free part is non-physical and violates the divergence condition. It also 50 resides in the null space of the operator A of (3.2.5). Therefore, an eigen solver which minimizes the Rayleigh quotient will inevitably converge to this null space. It is thus meaningful to include the divergence free constraint to eliminate the non-physical eigensolutions. A mixed formulation based on Lagrange multiplier p(∈ H01 (Ω) ≡ v ∈ H (Ω) | v |∂Ω = ) [5, 14] is a typical method to relax such a condition. The modified operators are such that,  ∇⊥ × ε−1 ∇⊥ × + β ε−1 iβε−1 ∇⊥ iβ∇⊥ · ε−1 −∇⊥ · ε−1 ∇⊥ ∇⊥ · iβ   A(ε, β) ≡   ∇⊥       iβ  , M ≡    H⊥       , u ≡  Hz   p (3.2.8)      I I  H (Ω) and H(curl, Ω) conforming basis We briefly review the two types of interpolation basis functions to be used for the finite element discretization of equation (3.2.8). Throughout our work, we employ isoparametric quadrilateral elements, i.e., the shape functions for geometric mapping from the reference coordinate system (ξ, η) to the (x, y) coordinate system are of the same order as the H (Ω) conforming nodal basis functions. η ( −1,1) τˆ3 (1,1) τˆ2 τˆ4 ξ ( −1, −1) τˆ1 (1, −1) Figure 3.14: Reference quadrilateral element. Over the reference square element shown in 3.14, recall that the linear H (Ω) conforming basis functions ϕˆi (ξ, η), i = 1, . . . , 4, can be expressed as (illustrated in Figure 3.15) ϕˆ1 = (1−ξ)(1−η), ϕˆ2 = (1+ξ)(1−η), ϕˆ3 = (1+ξ)(1+η), ˆ = (1−ξ)(1+η). ψ (3.2.9) 51 η η ϕˆ1 ϕˆ2 ξ ξ ϕˆ4 ϕˆ3 η η ξ ξ Figure 3.15: H (Ω) conforming basis functions. The hat denotes a quantity expressed relative to the reference coordinate system (ξ, η). The scalar field variable u ˆz over the element can be interpolated as u ˆz = uzi ϕˆi , (3.2.10) i=1 where the scalar coefficients uzi are the unknowns. ˆ i (ξ, η), i = Next, the lowest order H(curl, Ω) conforming basis functions ψ 1, . . . , 4, over the reference element can be expressed as   1 ˆ = (1−η)  , ψ   ˆ = (1+ξ)  , ψ   1 ˆ = (1+η)  , ψ   ˆ = (1−ξ)  . ψ (3.2.11) This is illustrated in Figure 3.16. If the unit tangent vector of an edge i is denoted by τi , then these vector basis ˆ j = δij , where δij is the Kronecker Delta. functions satisfy the requirements τi · ψ ˆ ⊥ over the element can be interpolated as Therefore, the vector field variable u   ˆξ   u ˆ⊥ =  u = u ˆη ˆ i, u⊥i ψ (3.2.12) i=1 where the scalar coefficients u⊥i are the unknowns and are constant on each edge i. 52 ψˆ1 ψˆ2 η η ξ ξ ψˆ3 ψˆ4 η η ξ ξ Figure 3.16: H(curl, Ω) conforming basis functions. Discretization We follow the similar procedure described in section 3.1.2 to discretize the infinite dimensional eigenvalue problem. First, we discretize the one-dimensional propagation constant in the range of [0, βM ]: Ph = {βt ∈ [0, βM ], ≤ t ≤ nβ } . (3.2.13) Second, the super cell Ωs is decomposed into N disjoint subcells Ki , ≤ i ≤ N , such that Ωs = ∪N i=1 Ki , and the shared interface of the two neighboring elements Ki and Kj is denoted by eij . The dielectric function ε(x, y, r) takes a uniform value between εL and εH on each element, where εL and εH are dielectric constants of a low-index material and a high-index material that make up the cross section of the photonic crystal fiber. That is, ε(r) = εj (∈ R) on Ki , and εL ≤ εj ≤ εH . Instead of the square and hexagonal lattices used in section 3.1, we use the rectangular and hexagonal lattices to avoid the double degeneracy of the fundamental space filling mode [38]. This choice is explained in details where the optimization problem is formulated in Chapter 5. As shown in Figure 3.17, both lattices have 4–fold symmetry. Dielectric variables are defined only on the highlighted cells, with the rest obtained through symmetry. 53 The symmetry lines are marked by red dashes. The structure of the crosssection can now be completely characterized by the dielectric variable vector for the cladding εCL , and the vector for the core εCO , i.e., ε = εCL ; εCO = CL CO CO εCL , . . . , εn CL , ε1 , . . . , εn CO . As before, the permissible set is defined as ε ε Qh ≡ {ε : ε ∈ [εL , εH ]nε }, (3.2.14) where nε = nεCL + nεCO . (a) Rectangular lattice (b) Hexagonal lattice Figure 3.17: Schematics of the cross-sections of photonic crystal fiber. Left: core region ΩCO made up of one primitive cell; middle: super cell Ωs ; right: one of the primitive cells Ω that make up of the cladding region. Third, the finite element “truth” approximation spaces of complex valued functions are defined as, Xh ≡ v ∈ H (Ω) | v|K ∈ P p (K), ∀K ∈ Th , (3.2.15a) Xh0 ≡ v ∈ H01 (Ω) | v|K ∈ P p (K), ∀K ∈ Th , (3.2.15b) Wh ≡ v ∈ H(curl, Ω) | v|K ∈ [P m (K)]2 , ∀K ∈ Th . (3.2.15c) P m is the space of complex valued polynomials of total degree at most m = mRE + mIM on element K. The weak formulation of the eigenproblem can be stated as follows, find (λ, u⊥ , uz , p) ∈ R × Wh × Xh × Xh0 , ∀(v⊥ , vz , q) ∈ 54 Wh × Xh0 × Xh0 , such that (∇⊥ × v⊥ , ε−1 ∇⊥ × u⊥ ) + β (v⊥ , ε−1 u⊥ ) + iβ(v⊥ , ε−1 ∇⊥ uz ) + (v⊥ , ∇⊥ p) = λ(v⊥ , u⊥ ), −iβ(∇⊥ vz , ε−1 u⊥ ) + (∇⊥ vz , ε−1 ∇⊥ uz ) + iβ(vz , p) = λ(vz , uz ), −(∇⊥ q, u⊥ ) + iβ(q, uz ) = 0. (3.2.16) Finally, we obtain the discrete equation m m Ah (ε, β)uh j = λmj Mh uh j , ε ∈ Qh , β ∈ Ph , (3.2.17) where    Ah (ε, β) =   A1 (ε) + β A2 (ε) iβB(ε) −iβB(ε)∗ D(ε) −C ∗ iβMz     iβMz     Mh =   C  M⊥     Mz    uh =  Hzh  ph (3.2.18) We can further rewrite the stiffness matrix in terms of its ε−dependence Ah (ε, β) = Ah (εCL , β) + Ah (εCO , β) + Ah (β) nεCL nεCO εCL i Ah,i (β) = i=1 (3.2.19) εCO i Ah,nεCL +i (β) + Ah,0 (β). + H⊥h i=1 Mesh refinement The refinement criteria and the treatment of the hanging nodes are exactly the same as described in section 3.1.3 for two-dimensional photonic crystal. The additional complications arises due to edge elements, i.e., H(curl) conforming bases. In our implementation, the lowest of the H(curl) bases are used, along each one of which the field variables take constant values. This leads to minimal modifications to the algorithm: when a hanging node splits an edge (e1 ) into two (e2 and e3 ), the field variables along these two edges e2 and e3 , each belonging to a different smaller element, and e1 , belonging to the larger neighboring element, should have the same value. Hence, these three edges should be assigned with one single degree of freedom. 55    .  3.2.3 Results and discussion Cladding eigenvalue convergence We first examined the convergence of the mixed formulation in both homogeneous and inhomogeneous domains, using the model problem where the material is homogeneous with dielectric constant ε(r) = everywhere. The analytical eigenvalue in this case is known to be λj = (mπ)2 + (nπ)2 + β , m, n ∈ Z on a computation domain Ω = [−1, 1]2 . Note that this is for the perfect cladding case, and the computation domain consists of one primitive unit cell. Quadrilateral meshes of both uniform and non-uniform sizes are considered. Shown in Figure 3.8, the uniform mesh is obtained by splitting a regular n × n Cartesian grid into a total of n2 squares, giving an uniform element size of h = a/n; The non-uniform mesh is obtained by refining half of the elements of the uniform mesh, giving the smallest element size of hmin = a/(2n). Three different meshes are used in each case, n = 8, 16, 32. The errors and convergence rate are defined as before in (3.1.12) and (3.1.13). Shown in Figure 3.18 is the convergence of the first 20 eigenvalues at several propagation constants (β = 0, π, 2π). The first observation is that the multiplicity of the eigenvalues are well captured by the uniform mesh, because only a few overlapped lines are visible in Figure 3.18(a); whereas in the nonuniform meshes, the degenerate eigenvalues deviate from each other by small margins. The convergence rate in both cases are also approximated to be r ≈ = min(p, m), where p and m are the orders of the polynomials employed for H ( Ω) and H(curl; Ω) conforming basis functions. The convergence of the first 20 eigenvalues in inhomogeneous domain is shown in Figure 3.19, with the same setup as in section 3.1.4, and Figure 3.10. Overall, a very stable convergence rate of r ≈ is demonstrated through the uniform meshes. In the non-uniform meshes, the convergence rate can be as high as r ≈ for the lower eigenmodes, and less satisfactory convergence (r < 2) for the higher modes, as the higher order eigenfunctions are not as sufficiently resolved. 56 β= β= −1 −1 10 10 equiv eh eh −3 10 10 −5 −5 10 −3 −2 10 −1 10 h 10 10 −2 10 −1 −1 10 equiv −3 eh eh 10 10 −3 10 −5 10 −5 −2 10 −1 10 h 10 −2 10 −1 10 h equiv β = 2π 10 β = 2π −1 −1 10 eh eh equiv 10 −3 10 −3 10 −5 10 −5 10 10 β= π β =π 10 −1 10 h equiv −2 10 −1 10 h (a) Uniform mesh 10 −2 10 −1 10 h equiv 10 (b) Non-uniform mesh Figure 3.18: Eigenvalue convergence on homogeneous domain at different propagation constants, β = 0, π, 2π. Each line represents the convergence of one of the first 20 eigenvalues, while the lowest line corresponds to the smallest eigenvalue. 57 β= β= −1 −1 10 10 equiv eh eh −3 10 −3 10 −5 10 −5 10 −2 10 −1 10 h 10 −2 10 β= π −1 −1 equiv eh 10 −3 eh 10 −3 10 −5 −5 −2 10 −1 10 h 10 −2 10 10 10 β = 2π −1 −1 10 equiv 10 −3 eh eh −1 10 h equiv β = 2π 10 −3 10 −5 −5 10 h equiv 10 β= π 10 10 −1 10 −2 10 −1 10 h (a) Uniform mesh 10 10 −2 10 −1 10 h equiv 10 (b) Non-uniform mesh Figure 3.19: Eigenvalue convergence on inhomogeneous domain at different propagation constants, β = 0, π, 2π. Each line represents the convergence of one of the first 20 eigenvalues, while the lowest line corresponds to the smallest eigenvalue. 58 Band structure To lay the ground work for the optimization problem, we examine in this section the effect of a core being introduced to the otherwise perfect periodic cladding. As introduced in section 1.1.1, a higher εCO would pull more modes beneath G,1 the light line. Hence, we would expect the fundamental eigenmode (λW ) and h G,2 some higher order modes (say λW ) of the waveguide to be smaller than the h fundamental space-filling mode, or light line of the cladding (λCL,1 ). Waveguide h of both rectangular and rhombic lattices are considered in this section. Two dielectric materials are used to construct the waveguide, epoxy of εL = 1.52 = 2.25, and silicon carbide of εH = 2.652 = 7.02. An advantage of using these two materials is that they are both solid at room temperature, therefore it simplifies the issue of non-connectivity. In Figure 3.20(a)(left), a primitive cell Ω of the cladding is constructed as a rectangular lattice (pitch distances Λx a = 1.5a and Λy a = a ) of dielectric material εH with cylindrical holes (εL ) of radius 0.485a. The primitive cells is used as the building blocks to construct the super cell Ωs of the waveguide, shown in Figure 3.20(a)(right), not to scale. The core of the waveguide is made up of one Ω with the εL hole filled up by εH . The cladding consists of two rings of Ω surrounding the core. In Figure 3.20(b), a waveguide set on a rhombic lattice (pitch distance Λ = a) with elliptical holes of major axis length 0.485a and minor axis 0.194a is constructed in the same way. In both cases, the dielectric function is computed on a uniform mesh of size a/20, and extrapolated to adaptively refined meshes, as shown in the two subplots of Figure 3.21. The corresponding dispersion relations of the light line and the first two waveguide modes computed on different computation meshes are shown in Figure 3.22. From (i) – (iv), the difference between the waveguide modes and light lines are calculated on successively finer meshes. In all of the four subplots of either rectangular lattice (a) or rhombic lattice (b), the differences between the light line G,1 and the fundamental waveguide mode, λCL,1 − λW , asymptotically approach h h zero from a positive value as the frequency decreases (or as the wavelength increases). The positivity of the difference for all β ≥ indicates that the subtrahend is a guided mode, and its asymptocity makes the subtrahend also a 59 −1 −1.5 1.5 −5 −7.5 7.5 (a) Rectangular lattice −1 −5 12 16 (b) Rhombic lattice Figure 3.20: Computation meshes for primitive unit cell Ω of cladding (left), and super cell Ωs of the waveguide (right). The core is formed by removing one εL hole and filling it with the εH . The cladding consists of two rings of primitive unit cells. 60 (i) hmin = a / 20 (ii) h = a / 40 −1 −1.5 1.5 (iii) hmin = a / 80 −1 −1.5 1.5 (iv) hmin = a / 160 −1 −1.5 1.5 −1 −1.5 1.5 (a) Rectangular lattice (i) hmin = a / 20 −1 (ii) h = a / 40 (iii) h = a / 80 (iv) h = a / 160 −1 −1 −1 (b) Rhombic lattice Figure 3.21: Dielectric function is first defined on the coarsest uniform mesh (i), and extrapolated adaptively to finer meshes sequentially (ii) – (iv). 61 cut-off free mode. However, the differences between the light line and the second G,2 waveguide mode, λCL,1 − λW , only show nonnegative values for propagation h h constants β > βc (e.g., in Figure 3.22(a), the intersection with the horizontal axis is around βc a/2π = 0.5). This guided mode is known as a cut-off guided mode. Lee et al have proposed conditions for which these cut-off free and cut-off guided modes would arise [39]. Also shown in the same figure, the guidedness and the cut-off properties of the waveguide modes are not affected by the resolutions of the computation meshes. 3.3 Conclusions This chapter focused on the numerical solutions to the macroscopic Maxwell equations in the form of Hermitian eigenvalue problems. The numerical recipe chosen for this task is the finite element method, which surpasses others in performance due to its convenient geometrical representation, parameter affinity, and flexible mesh adaptivity. Two variations based on different physical problems are considered. The first type, two dimensional photonic crystal fiber, describes a dielectric structure that is discretely periodic in the xy-plane, and invariant in the z-axis. The second type, the photonic crystal fiber, is constructed by introducing perturbations across the xy-plane to the perfectly periodic 2D photonic crystal of the first type. This can also be treated as a two-dimensional structure due to its homogeneity along the z-axis. The governing equation of the two-dimensional photonic crystal can be simplified to two separate scalar equations depending on the polarization of the electromagnetic waves propagating in the xy-plane: transverse electric, or transverse magnetic. These two Laplace-like scalar eigenvalue equations can be discretized by standard finite element method using nodal basis functions for shape and field variable interpolations. The governing equation of the photonic crystal fiber has additional complexity as the propagation direction is along the z-axis, and therefore no advantage can be taken of the polarization. In addition, the special mixed method between edge basis and nodal basis was used for the finite 62 (i) h = a / 20 0.02 0.016 λ a 2/4 π2 λ a 2/4 π2 0.016 0.012 0.008 0.004 (ii) hmin = a / 40 0.02 0.008 0.004 CL,1 WG,1 λ− − λ −λ h CL,1 0.012 h WG,2 CL,1 WG,1 λ− − λ −λ h CL,1 0.5 1.5 0 0.5 (iii) h = a / 80 0.016 λ a 2/4 π2 0.012 0.008 CL,1 WG,1 λ− − λ −λ 0.004 h CL,1 0.012 0.008 CL,1 WG,1 λ− − λ −λ 0.004 h WG,2 h CL,1 0.5 1.5 h WG,2 λ− − λh − λh λ− − λh − λh 1.5 (iv) hmin = a / 160 0.02 0.016 λ a 2/4 π2 βa/2π βa/2π 0.02 h WG,2 λ− − λh − λh λ− − λh − λh 0.5 βa/2π 1.5 βa/2π (a) Rectangular lattice (i) h = a / 20 0.01 (ii) hmin = a / 40 0.01 CL,1 WG,1 λ− − λ −λ CL,1 WG,1 λ− − λh − λh CL,1 0.006 0.004 0.002 h CL,1 0.006 0.004 0.002 0.5 1.5 0.5 (iii) h = a / 80 0.01 h CL,1 0.004 0.002 h CL,1 h WG,2 λ− − λh − λh 0.008 λ a 2/4 π2 λ a 2/4 π2 CL,1 WG,1 λ− − λ −λ h WG,2 λ− − λh − λh 0.006 (iv) hmin = a / 160 0.01 CL,1 WG,1 λ− − λ −λ 0.008 1.5 βa/2π βa/2π h WG,2 λ− − λh − λh 0.008 λ a 2/4 π2 0.008 λ a 2/4 π2 WG,2 λ− − λh − λh 0.006 0.004 0.002 0.5 βa/2π 1.5 0.5 1.5 βa/2π (b) Rhombic lattice Figure 3.22: Dispersion relation of structures corresponding to those shown in Figures 3.20 and 3.21 on the coarsest uniform mesh (i), and to finer meshes sequentially (ii) – (iv). 63 element approximation. In either case, we end up with a generalized eigenvalue problem that can be solved by various standard eigen solvers; “eigs” from MATLAB is our solver of choice throughout this thesis. Mesh adaptivity has also been introduced and incorporated to discretize the computation domain non-uniformly and represent the material property more efficiently. Finally, we examined the convergence rate of the numerical methods on both uniform and non-uniform meshes, homogeneous and inhomogeneous domains of various lattices. A consistent convergence rate of r ≈ has been obtained for the eigenmodes in the uniform meshes. In the non-uniform meshes, the convergence rate can be as high as r ≈ for the lower eigenmodes. These validated results laid the ground work for the optimization problems in the next two chapters. 64 [...]... example A 12 7 5.5 Optimal PCF structures and the field intensity, example B 12 9 5.6 Optimal PCF structures and the field intensity, example C 13 0 5.7 Optimal PCF structures and the field intensity, example D 13 1 5.8 Optimal PCF structures and the field intensity, example E 13 2 xiii Chapter 1 Introduction 1. 1 1. 1 .1 Background Photonic crystals Photonic crystals will grant us the control of light... 10 3 4.27 Optimization results for multiple complete band gaps in the square lattice 10 3 xii 5 .1 Example of dispersion relation of a single-polarization single-mode photonic crystal fiber 10 6 5.2 Formulation II of the optimal design of SPSM PCF 11 5 5.3 Evolution of the optimization process based on PIc 12 5 5.4 Optimal PCF...List of Figures 1. 1 Schematic examples of one-, two-, and three-dimensional photonic crystals 1. 2 2 Schematic illustrations of various waveguide operating with the index guiding mechanism 3 1. 3 Example of dispersion relation of a photonic crystal fiber 4 1. 4 Field profiles of localized and non-localized modes 5 3 .1 Square and hexagonal... notion of band width, similar to the band gap discussed before, is defined as a range of frequency at certain wave vector(s) in which only a single guided mode exists 1. 1.2 Optimal design The optimal conditions for the appearance of photonic band gaps were first studied for one-dimensional crystals by Lord Rayleigh in 18 87 [48] A one dimensional photonic crystal consists of alternating layers of material... prevalent 1. 2 1. 2 .1 Scope Thesis contributions The central theme of the work in this thesis is the optimal design of photonic crystal using convex optimization We propose a new approach based on semidefinite programming (SDP) and subspace methods for the optimal design of photonic band structure In the last two decades, SDP has emerged as the most important class of models in convex optimization, [1, 3,... (2 .11 ) A periodic boundary condition is imposed on the boundary of Ω, denoted by Γ, to align with our physical problems later To derive the weak form of the governing equation, we first introduce a function space X e = {v ∈ H 1 (Ω)}, (2 .12 ) where H 1 (Ω) = {v ∈ L2 (Ω)| v ∈ (L2 (Ω))d } (2 .13 ) 21 The associated norm is defined as 1/ 2 d v Xe = vi 2 H 1 (Ω) , (2 .14 ) i =1 where d indicates the dimension of. .. frequencies from 1 Figure 1. 1: Schematic examples of one-, two-, and three-dimensional photonic crystals The dimensionality of a photonic crystal is defined by the periodicity of the dielectric materials along one or more axes propagating in certain directions Moreover, photonic crystals can channel propagation of light in more effective ways than homogeneous dielectric media, such as index guiding in photonic. .. 46 3 .13 Band diagrams on a hexagonal lattice 47 3 .14 Reference quadrilateral element 51 3 .15 H 1 (Ω) conforming basis functions 52 3 .16 H(curl, Ω) conforming basis functions 53 3.9 x 3 .17 Discretization of photonic crystal fiber 54 3 .18 Eigenvalue convergence on homogeneous domain 57 3 .19 Eigenvalue convergence... improve the computational efficiency Extensive optimal designs of the two-dimensional photonic crystals are presented with optimal band gaps of various configurations, e.g., absolute band gaps, complete band gaps, and multiple band gap In chapter 5, we study the band width optimization problem arising in the photonic crystal fiber, and investigate the design of the single-mode single polarization fibers... (v) The set of all linear functionals on a linear space X is itself a linear space This space, denoted by X , is called the dual space of X Bilinear functional Let X and Y be two linear spaces over the field F An operator a : X × Y → F is a bilinear form if and only if, ∀u1 , u2 ∈ X, v1 , v2 ∈ Y , and α, β, γ, λ ∈ F ¯ ¯ a(αu1 + βu2 , γv1 + λv2 ) = α¯ a(u1 , v1 ) + αλa(u1 , v2 ) + β¯ a(u2 , v1 ) + β λa(u2 . E. . . . 13 2 xiii Chapter 1 Introduction 1. 1 Background 1. 1 .1 Photonic crystals Photonic crystals will grant us the control of light. When properly designed, these structures, comprised of periodically. . . . . . . . . vii List of Tables ix List of Figures xiii 1 Introduction 1 1 .1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 .1. 1 Photonic crystals . . . . . . . relation of a single-polarization single-mode photonic crystal fiber. . . . . . . . . . . . . . . . . . . . . . . . . 10 6 5.2 Formulation II of the optimal design of SPSM PCF. . . . . . . . 11 5 5.3